Reversible Cyclic Voltammetry and Non-Unity Stoichiometry: The Ag/AgBr/Br– Redox Couple

The voltammetry of electrochemically reversible couples in which a soluble reactant is converted into an insoluble product is investigated computationally via simulation and, in the context of the Ag/AgBr/Br–redox couple, experimentally. The voltammetric waveshape is characterized and, when analyzed via apparent transfer coefficient analysis, shown to give rise to apparent transfer coefficients very considerably in excess of unity, leading to the generic insight for the characterization of electrode reactions involving solution and solid phase reactants.

First, the voltammogram is converted to dimensionless form using the table shown in the text except for the definition of the dimensionless potential which is noted in the main text. The dimensionless voltammogram then takes the form of dimensionless flux, J, versus dimensionless potential. where and k represents the current spatial and time steps.
The equation can be rearranged as: where From the boundary condition it can further be inferred that = Δ -= 0 = Δ Using the Backward Implicit method, a tridiagonal sparse matrix is constructed to solve for the concentration profile at time step : where , , . [ By solving the concentration profile sequentially for increasing k, the surface concentration as a function of time or potential can be obtained.
To validate our implementation of the inversible method prior to application to the (0,1) system of interest, the experimental voltammogram for the one electron reduction of to   Simulation of the electrochemical reaction model utilized expanding space grids 3 to reduce computational time. If the scans start and finish at the same potential, the time for the full scan is , where and are the dimensionless starting and switching potentials = respectively. is the dimensionless scan rate. The maximum spatial distance of the simulation in solution is taken as 4 . The accuracy of the simulations, were dependent on the spatial = 6 step size and time step size. The spatial step size in an expanding grid depends on the initial step size, , and the expanding grid factor, , according to the following definition: ℎ 0 where is the dimensionless spatial coordinate at point . In simulation, ℎ = + 1 -= ℎ ℎ 0 was related to by where is a simulation factor determining the initial spatial step. ℎ 0 = To test convergence of the simulation, was varied from to and varied from 1.01 to 1.2. 10 -9 10 -5 Since a uniform time grid was used, was varied from to while was fixed at 1600 and 10 -5 10 -2 100, the highest and lowest scan rates presented in the paper.
The voltammograms generated were validated by comparing the forward scan peak flux, and peak potential, at different values of and as shown in Figure  To clarify the solution conditions relating to the voltammetry reported in the main text, the speciation of silver (I) in the solutions of interest were calculated using Hydra/Medusa. 5 The chemical equilibria considered were as follows:  Table S 1. For voltammetry in a 1.6 mM bromide solution, AgBr(aq) accounts for 76% of the silver (I), while AgBr 2and Ag + are 11% and 13% respectively. Note that is not present 2 -3 in solutions with low concentration of bromide 6 .  By integrating the oxidative and reductive charges the fraction of stripped AgBr was obtained as a function of and scan rates are shown in Table S 2. The fraction stripped decreases with a decrease *in scan rates and decrease in bulk concentration of bromide. The trends are discussed in the main text.

AgBr deposit thickness
To estimate the AgBr deposit thickness during the experimental voltammetric scans, the current was integrated versus time to obtain the charge transferred, which was then used to estimate the AgBr thickness on the electrode using the following equation: The parameters used to calculate AgBr thickness are shown in Table S 3. Figure S 5 gives an example of the estimated AgBr deposit thickness as the voltammetric scan proceeds. The summary of the peak AgBr thickness at different bromide bulk concentrations and scan rates are shown in Table S 4.